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# Beta decay

Nuclear physics
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In nuclear physics, beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted. In the case of electron emission, it is referred to as "beta minus" (β), while in the case of a positron emission as "beta plus" (β+).

In β decay, the weak interaction converts a neutron (n0) into a proton (p+) while emitting an electron (e) and an anti-neutrino ($\bar{\nu}_e$):

$n^0 \rightarrow p^+ + e^- + \bar{\nu}_e$.

At the fundamental level (as depicted in the Feynman diagram below), this is due to the conversion of a down quark to an up quark by emission of a W- boson; the W- boson subsequently decays into an electron and an anti-neutrino.

In β+ decay, energy is used to convert a proton into a neutron, a positron (e+ ) and a neutrino (νe):

$\mathrm{energy} + p^+ \rightarrow n^0 + e^+ + {\nu}_e$.

So, unlike beta minus decay, beta plus decay cannot occur in isolation, because it requires energy, the mass of the neutron being greater than the mass of the proton. Beta plus decay can only happen inside nuclei when the absolute value of the binding energy of the daughter nucleus is higher than that of the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron and a neutrino and into the kinetic energy of these particles.

In all the cases where β+ decay is allowed energetically (and the proton is a part of a nucleus with electron shells), it is accompanied by the electron capture process, when an atomic electron is captured by a nucleus with the emission of a neutrino:

$\mathrm{energy} + p^+ + e^- \rightarrow n^0 + {\nu}_e$.

But if the energy difference between initial and final states is low (less than 2mec2), then β+ decay is not energetically possible, and electron capture is the sole decay mode.

If the proton and neutron are part of an atomic nucleus, these decay processes transmute one chemical element into another. For example:

$\mathrm{{}^1{}^{37}_{55}Cs}\rightarrow\mathrm{{}^1{}^{37}_{56}Ba}+ e^- + \bar{\nu}_e$ (beta minus),
$\mathrm{~^{22}_{11}Na}\rightarrow\mathrm{~^{22}_{10}Ne} + e^+ + {\nu}_e$ (beta plus),
$\mathrm{~^{22}_{11}Na} + e^- \rightarrow\mathrm{~^{22}_{10}Ne} + {\nu}_e$ (electron capture).

Beta decay does not change the number of nucleons A in the nucleus but changes only its charge Z. Thus the set of all nuclides with the same A can be introduced; these isobaric nuclides may turn into each other via beta decay. Among them, several nuclides (at least one) are beta stable, because they present local minima of the mass excess: if such a nucleus has (A, Z) numbers, the neighbour nuclei (A, Z−1) and (A, Z+1) have higher mass excess and can beta decay into (A, Z), but not vice versa. It should be noted, that a beta-stable nucleus may undergo other kinds of radioactive decay (alpha decay, for example). In nature, most isotopes are beta stable, but a few exceptions exist with half-lives so long that they have not had enough time to decay since the moment of their nucleosynthesis. One example is 40K, which undergoes all three types of beta decay (beta minus, beta plus and electron capture) with half life of 1.277×109 years.

Some nuclei can undergo double beta decay (ββ decay) where the charge of the nucleus changes by two units. In most practically interesting cases, single beta decay is energetically forbidden for such nuclei, because when β and ββ decays are both allowed, the probability of β decay is (usually) much higher, preventing investigations of very rare ββ decays. Thus, ββ decay is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change A; thus, at least one of the nuclides with some given A has to be stable with regard to both single and double beta decay.

Beta decay can be considered as a perturbation as described in quantum mechanics, and thus follows Fermi's Golden Rule.

## Kurie plot

A Kurie plot (also known as a Fermi-Kurie plot) is a graph used in studying beta decay, in which the square root of the number of beta particles whose momenta (or energy) lie within a certain narrow range, divided by a function worked out by Fermi, is plotted against beta-particle energy; it is a straight line for allowed transitions and some forbidden transitions, in accord with the Fermi beta-decay theory.

### References

• Franz N. D. Kurie, J. R. Richardson, H. C. Paxton (March 1936). "The Radiations Emitted from Artificially Produced Radioactive Substances. I. The Upper Limits and Shapes of the β-Ray Spectra from Several Elements". Physical Review 49 (5): 368-381. doi:10.1103/PhysRev.49.368.
• F. N. D. Kurie (May 1948). "On the Use of the Kurie Plot". Physical Review 73 (10): 1207. doi:10.1103/PhysRev.73.1207.

## History

Historically, the study of beta decay provided the first physical evidence of the neutrino. In 1911 Lise Meitner and Otto Hahn performed an experiment that showed that the energies of electrons emitted by beta decay had a continuous rather than discrete spectrum. This was in apparent contradiction to the law of conservation of energy, as it appeared that energy was lost in the beta decay process. A second problem was that the spin of the Nitrogen-14 atom was 1, in contradiction to the Rutherford prediction of ½.

In 1920-1927, Charles Drummond Ellis (along with James Chadwick and colleagues) established clearly that the beta decay spectrum is really continuous, ending all controversies.

In a famous letter written in 1930 Wolfgang Pauli suggested that in addition to electrons and protons atoms also contained an extremely light neutral particle which he called the neutron. He suggested that this "neutron" was also emitted during beta decay and had simply not yet been observed. In 1931 Enrico Fermi renamed Pauli's "neutron" to neutrino, and in 1934 Fermi published a very successful model of beta decay in which neutrinos were produced.

Making it simple to understand the concept of beta decay is generally represented in the following way:

$\mathrm{~^{A}_{Z}X}_{N}\rightarrow\mathrm{~^{A}_{Z+1}Y}_{N-1} + e^- + \bar{\nu}_e$ (beta minus)
$\mathrm{~^{A}_{Z}X}_N\rightarrow\mathrm{~^{A}_{Z-1}Y}_{N+1} + e^+ + {\nu}_e$ (beta plus)
$\mathrm{~^{A}_{Z}X}_N+ e^-\rightarrow\mathrm{~^{A}_{Z-1}Y}_{N+1} + {\nu}_e$ (electron capture)

Where X and Y represent the parent and daughter nuclei respectively, (A= mass number, Z= atomic number, N= number of neutrons).